In Chapter IV, we start to explore the connection between randomness and games. A more systematic study is made of the probabilistic approach, that is actually refered to as a “fake probabilistic method.”

The main ingredients of the “fake probabilistic method” are:

1. (1) the two linear criterions (“Part A”) – for some applications see Part B;

2. (2) the advanced Weak Win criterion together with the ad hoc method of Section 23 (“Part C”);

3. (3) the BigGame–SmallGame Decomposition and its variants (“Part D”).

The main result in Chapter V is (2): the Advanced Weak Win Criterion, a complicated “higher moment” criterion. It is complicated in many different ways:

1. (i) the form of the criterion is already rather complicated;

2. (ii) the proof of the criterion is long and complicated;

3. (iii) the application to the Clique Game requires complicated calculations.

This criterion basically solves the building part of the Meta-Conjecture (see Section 9).

Motivating the probabilistic approach

Let us return to Section 6: consider the Maker-Breaker version of the (KN, Kq) Clique Game (we don't use the notation [KN, Kq] any more). How do we prove lower bound (6.1)? How can Maker build such a large clique?

Halving Argument. The Ramsey criterion Theorem 6.2, combined with the Erdős–Szekeres bound, gives the size q = ½ log2N, which is roughly ¼ of the truth.